Transcript No Slide Title
Dominic F.G. Gallagher Thomas P. Felici
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Outline EME = “
E
igen
m
ode
E
xpansion”
•Introduction to the EME method - basic theory - stepped structures - smoothly varying - periodic •Why use EME?
Comparison to BPM and FDTD •Examples
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A
modes:
E
(
x
,
y
,
z
)
e m
(
x
,
y
)
e i
m z
B
“
The fields at AB of any solution of Maxwell’s Equations may be written as a superposition of the modes of cross section AB”.
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Basic Theory
propagation constant
E
(
H
(
x
,
x
,
y
,
y
,
z
)
z
)
m
m
E H
m
(
x
,
m
(
x
,
y
)
y
) (
c m
.
e i
m
(
c m
.
e i
m z
z
c m
.
e
i
m c m
.
e
i
m z
)
z
) electric field magnetic field forward amplitude backward amplitude mode profiles •exact solution of Maxwell’s Equations •bi-directional
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•so far only z-invariant •what about a step change?
a m (+) a m (-) b b m m (+) (-)
Maxwell's Equations gives us continuity conditions for the fields, e.g. the tangential electric fields must be equal on each side of the interface
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•Applying continuity relationships, eg: LHS
k N
1
a
forward ( )
e k
i
k z
backward
a k
( )
e
i
k z
.
E
(
a
k , t ) (
x
.
y
)
k N
1
b k
( )
e i
k z
forward
b k
( )
e
i
k z
.
E
(b) k, t (
x
.
y
) backward RHS With orthogonality relationships and a little maths we get an expression of the form:
a
( )
b
( )
S J a
(
b
( ) ) S J is the
scattering matrix
of the joint
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A
Straight Waveguide
B trivial - the
scattering matrix
is diagonal:
S AB
e i
1
z
0 0 0
e i
2
z
0 0 0
e i
3
z
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Example: S-matrix decomposition of an MMI coupler
A B C D E
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A Evaluating S-matrix of device B C D E AB C DE ABC DE ABCDE
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Smoothly Varying Elements • Problem: modes are changing continuously along element • Thus each cross-section requires a large computational effort to locate the set of modes • This was the major hurdle that has in the past restricted application of EME • FIMMPROP (our implementation of EME) has tackled these problems, enabling EME to be used realistically for the first time even for 3D tapering structures.
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h n 0 Order (Staircase Approximation) h n Set of local modes computed at discrete positions along taper • Simple to implement • theoretically accurate as Nslice • errors grow as Nslice so practical limit on Nslice • can get spurious resonances between modes for long structures and small Nslice
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h n 1st Order (Linear Approximation) analytic integration h n Set of local modes computed at discrete positions along taper • More complex to implement • good accuracy for modest Nslice • errors 0 for modest Nslice • need only small number of modes
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Showing zero order versus first order result 1 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0 100 200 300
taper length (um)
400 500
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Periodic Structures A B A B A B A B A B P1 P2 P1 P1 P1 P1 P2 P3 S • compute time log(Nperiod) • i.e. almost as quick as a straight waveguide!
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transmission = (Sj) N periodic structure Sj Bends • bend is just periodic repeat of straight waveguide sections!
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Boundary Conditions •
PEC
&
PMC
(perfect electric/magnetic conductors) - useful for exploiting symmetries •
transparent
boundary conditions •
PML
’s - perfect matched layers (with PEC or PMC) • Transparent BC’s are naturally formed at input and output of EME computation • finding eigenmodes with true transparent boundary conditions leads to
leaky modes
- leaky modes cause problems with completeness of basis set.
• PML much better suited for finding modes for EME than leaky modes -
completeness
better achieved.
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PEC The Perfect Matched Layer (PML) d1 (real) d2=a+jb PEC waveguide core/cladding PML • Imaginary thickness of PML absorbs light
propagating
towards boundary • as much absorption as we wish with no reflection at cladding/PML interface!
• guided modes not absorbed at all - nice!
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Effect of PML on guided and radiating modes guided mode unbound mode PML core
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PML
PML’s with segmented waveguide
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Why Use EME?
EME Advantages 1. rigorous solution of Maxwell's Equations - rigorous as Nmode infinity - large delta-n
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2. inherently bi-directional. - unconditionally stable since always express (outputs) = S.(inputs) - takes any number of reflections into account - not iterative - even highly resonant cavities - mirror coatings, multi-layer
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EME Advantages 3. The S-matrix technique provides the solution for all inputs!
- component-like framework where the S-matrix of one component may be re used in many different contexts.
Other methods: •Input 1 calculate •Input 2 calculate •Input 3 calculate Result 1 Result 2 Result 3 EME/FIMMPROP: •Calculate S matrix •Input 1 Result 1 •Input 2 Result 2 •Input 3 Result 3
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EME Advantages 4. Hierarchical algorithm permits re-use, accelerating computation of sets of similar structures. When one part of a device is altered only the S-matrix of that part needs to be re-computed. • Initial evaluation time: ~ 2:54 m:s • change period - time: …… • change offset - time: ….
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Design Curve Generation
Traditional Tool: 5 mins 5 mins 5 mins EME/FIMMPROP: 5 mins
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3 mins 5 mins 5 mins 5 mins
EME Advantages 5. Wide-angle capability - wider angle - just add more modes - adapts to problem
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EME Advantages 6. The optical resolution and the structure resolution may be different.
- c.f. BPM (stability problems with non-uniform grid) • very thin layers - wide range of dimensions • no problem for EME/FIMMPROP algorithm does not need to discretise the structure
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Plasmonics Right: plasmon between silver plates EME is a rigorous Maxwell solution and can model many plasmonic devices (provided basis set is not too large).
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Why Use EME?
Disadvantages • Structures with very large cross-section are less suitable for the method since computational time typically scales in a cubic fashion with e.g. cross-section width.
• The algorithms are much more complex to write - for example it is very difficult to ensure that a mode has not been missed from the basis set.
• EME is not a "black box" technique - the operator must make some effort to understand the method to use it to his best advantage.
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Computation Time • Compare computation time with BPM and FDTD • Restrict our discussion to 2D - i.e. z and one lateral dimension • Both BPM and FDTD require a finite difference grid sampling the structure, this same grid used for optical field • EME does not need a structure grid (FMM Solver) • Equivalent of grid in EME is the number of modes • For straight wg, EME particularly efficient • Periodic section - logarithmic time • Arbitrary z-variations - all 3 methods have similar dependence with z complexity/resolution • In lateral dimension EME gets high spatial resolution almost for free. (c.f. BPM, non uniform grid problems…) • But lateral optical resolution - compute time N 3 the
limiting factor
in EME
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Table1: Computation time – dependence on spatial resolution (2D simulation)
Straight waveguide – z dependence Near adiabatic structure – e.g. taper Periodic structure – z dependence Generic structure – z dependence Cross-section – structural resolution Cross-section – optical resolution BPM N 1 N 1 N 1 N 1 N 1 N 1 FDTD N 1 N 1 N 1 N 1 N 1 N 1 EME N 0 N 0 N 1 log(N) N 1 . N 1 , small N 3
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Memory Requirements • Very efficient as a function of z-resolution - N 0 scaling for straight or periodic
Table 2: Memory requirements – dependence on spatial resolution (2D simulation)
Straight waveguide – z dependence Periodic structure – z dependence Generic structure – z dependence Cross-section – structural resolution Cross-section – optical resolution BPM N 0 N 1 N 0 N 1 N 0 N 1 N 1 N 1 FDTD N N 1 N 1 N 1 N 1 1 EME N N 0 N 1 N 1 N 0 0 N N 1 N 2 1
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Applications • We present a variety of examples illustrating EME features
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SOI Waveguide Modes Ex Field
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Ey field
The MMI • MMI ideal for EME - inherently a modal phenomenon • 8 modes • Illustrate design scan - 100 simulations for price of two!
1 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0
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0.5
1 1.5
input offset (um)
2 2.5
3
Tapered Fibre • 6 modes • “how long for 98% efficiency?” - ideal question for EME
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0.4
0.3
0.2
0.1
0 1 0.9
0.8
0.7
0.6
0.5
0 2000 4000 6000 taper length ( m m) 8000 10000 • 50 simulations in scan • 6.5s per simulation (in 3D!)
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Effective indices of first 5 modes 1.472
1.470
1.468
1.466
1.464
1.462
1.460
1.458
1.456
0 10 20 30 40 50 60 70 80 90 100 • Strong coupling occurs when the effective indices are close - telling us where the trouble spots of the device are.
• Powerful diagnostic - tells designer
where
to
Co-directional Coupler • remember logarithmic with N periods • very thin layer - no problem
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propagation at sub wavelength scales, including metal features
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Ring Resonator • Nmodes = 60 (for one ring) • Nmodes much higher here - wide angle propagation.
• BPM gives nonsense
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Photonic Crystal Design • Nmodes = 60 (for one ring) • Nmodes much higher here - wide angle propagation.
• BPM gives nonsense
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VCSEL Modelling top DBR mirror active layer lower DBR mirror • Resonance Condition
R top
.
R bot u
g u
R top R bot
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Silicon nanowires – high NA 2D 3D DBR Gratings EME n FDTD Small NA BPM Fibre Taper Device length Comparison of computational domains of EME, BPM and Showing the domains of applicability of FDTD, BPM and EME to varying delta-n and device length.
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DBR Gratings FDTD EME BPM 3D 3D AWG cross-section size Showing the domains of applicability of FDTD, BPM and EME to varying numerical apperture and cross-section size.
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BPM – Capability Scores
Aspect
Speed Memory NA
Performance
FD-BPM scales linearly with area and can take fairly long steps in propagation direction -
Score/10
Usage scales linearly with c/s area Best with low NA simulations. Versions using Pade approximants can model a beam travelling at a large angle but still cannot deal well with light simultaneously travelling at a wide range of angles.
4 5 5 Delta-n Polarisation Lossy materials Reflections Non-linearity Dispersive Geometries Best with low delta-n simulations. Semi-vectorial versions work best. Still problems modelling mixed or rotating polarisation structures accurately Can model modest losses efficiently. Most versions cannot deal well with metals Some success in implementing reflecting/bi-directional BPM but generally avoided due to slow speed and stability problems FD-BPM can model non-linearity.
Being a frequency-domain algorithm this is easy The BPM grid allows diffuse structures to be modelled easily. Problems modelling non-rectangular structures accurately on the rectangular grid 7 3 9 10 7 ABCs PMLs available and work well 9
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Aspect
Speed Memory NA Delta-n Polarisation Lossy materials Reflections Non-linearity Dispersive Geometries ABCs FDTD – Capability Scores
Performance
Scales as V (device volume) but grid size is small so not as good as BPM or EME for long devices.
Scales as V (device volume) but grid size is small so not as good as BPM or EME for long devices.
Omni-directional algorithm is agnostic to direction of light – great when light is travelling in all directions Rigorous Maxwell solver, happy with high delta-n, but slows down somewhat with high index.
Rigorous Maxwell Solver is polarisation agnostic Can model even metals accurately with a fine enough grid and small modifications to the algorithm.
Yes – easy and stable even when there are many reflecting interfaces.
-
Score/10
10 9 10 10 Yes – non-linear algorithm relatively easy to do Have to approximate the dispersion spectrum with one or more Lorentizans but exact fit to the spectrum over a wide wavelength is difficult and the algorithm also slows down.
Fine rectangular grid can do arbitrary geometries easily, though there are problems approximating diagonal metal surfaces Yes – very effective and easy to use 9 7 8 9
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Aspect
Speed Memory NA Delta-n Polarisation Lossy materials Reflections Non-linearity Dispersive Geometries ABCs EME – Capability Scores
Performance
EME scales poorly with cross-section area – as A 3 (A is c/s area). However it can efficiently model very long structures especially if their cross-section changes only slowly or occasionally. Periodic structures scale as log(number of periods) – so can compute efficiently. S-matrix approach allows a set of similar simulations to be done very quickly – parts of previous calculation can be reused.
Memory increases at rate between A 2 and A 3 (A is c/s area), but very efficient for long or periodic devices.
Can model wide-angle beams by increasing the number of modes in the basis set at expense of speed and memory.
Rigorous Maxwell Solver can accurately model high delta-n Rigorous Maxwell Solver is polarisation agnostic Depends on mode solver used. -
Score/10
7 8 10 7 Yes – easy and stable even when there are many reflecting interfaces.
10 Difficult – have to iterate, and then only modest non-linearity levels will converge Being a frequency-domain algorithm this is easy Depends on the mode solver used. Can use different structure discretisations in different cross-sections, so solver can better adapt to the geometry. Depends on the mode solver used. E.g. a finite-difference solver can be readily constructed to implement PMLs. However, PML’s are more difficult to use with EME than with BPM or FDTD.
3 10 7 7
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Conclusions • Powerful compliment to BPM and FDTD • Exceedingly efficient/fast for wide range of examples • Rigorous Maxwell solver, bi-directional, wide angle • mode data provides important insight into workings of device.